Critical points, tangent line, graphing

1) Consider . Find all values, c, in the interval [0,1] such that the slope of the tangent line to the graph of f at c is parallel to the secant line through the points (0, f(0)) and (1, f(1)).

2) Use a graphing calculator to graph the function .
a) Adjust the viewing window and use the zoom and trace features to estimate the x-values of the relative extrema to one decimal place.
b) Use calculus to find the actual x-values of the relative extrema.

3) Use a graphing calculator to graph the function .
a) Use the graph to determine open intervals of increasing or decreasing.
b) Use calculus to find the actual intervals of increase and decrease.

4) Use a graphing calculator to graph .
a) Use the graph to determine the open intervals where the graph of the function is concave upward or concave downward.
b) Use calculus to find the actual intervals of concavity.

5) Find the limits:
a) b) c)
6) Consider .
a) Find all asymptotes.
b) Use a graphing calculator to graph f.
c) Use the graph to find the point on the graph where the curve crosses the horizontal asymptote.

7) An open box is to be made from a rectangular piece of cardboard, 7 inches by 3 inches, by cutting equal squares from each corner and turning up the sides.
a) Write the volume, V, as a function of the edge of the square, x, cut from each corner.
b) Use a graphing calculator to graph the function, V. Then use the graph of the function to estimate the size of the square that should be cut from each corner and the volume of the largest such box.

8) Use a graphing calculator to graph .
a) Use the graph to estimate (to one decimal place) the real zero of f.
b) Approximate this zero using the value found in part a and Newton's Method until two successive approximations differ by less than 0.001.

9) Consider .

a) Find an equation of the tangent line, T, at the point (2,8).
b) Graph f and T on the same coordinate axes using a graphing calculator.
c) Use the graphs to estimate f(2.1) and T(2.1).
d) Calculate the actual values of f(2.1) and T(2.1).